Formulated the laws of classical logic. Laws of logic (4) - Law

Laws of logic(or logical laws) is the general name for the set of laws that form the basis logical deduction(cm. ). The concept of a logical law goes back to the ancient concept of logos(see) as a prerequisite for objective (“natural”) correctness of reasoning. Because the logics(see) studies the nature of the connection of thoughts in the process of reasoning, there are certain formal And meaningful rules that must be followed. Different in structure and degree of complexity, reasoning obeys different rules. Among them are main And derivatives: basic rules have more general character, derivatives are derived from the main ones. Along with this, there is a type of rules of logic that can be called universal. These rules are usually called laws of thought. By law, in general, they mean the internal, necessary and essential connection of phenomena. The laws of thought are operational directives of thinking. Their origin is due to the rational activity of the subject. Expedient activity, expressed in rules, norms, recommendations, is embodied in principles that have a universal character. Unlike the laws of natural science, which describe the connection between natural phenomena, repeated many times under identical conditions, the laws of thinking prescribe certain methods of intellectual activity. The purpose of the laws of logic is to formulate the foundations of rules and recommendations, following which one can reach the truth. Therefore, the laws of thought are not laws in the sense in which the term is used to describe natural phenomena. Thus, the laws of logic are the laws of a person's correct thinking about the world, and not the laws of the world itself.

The rules of thinking for the first time receive logical content from Aristotle, who laid the foundation for the systematic description and cataloging of such schemes of logical connections of elementary statements into complex ones, the truth of which follows from their form alone, or rather, from the mere understanding of the meaning of logical connections, regardless of the truth value of elementary statements. . Most of the logical laws discovered by Aristotle are laws syllogism. Later, other laws were discovered, and it was even established that the totality of the laws of logic is infinite. In a sense, it is possible to consider this set with the help of various formal theories of logical reasoning - the so-called logical calculus, in which the intuitive concept of "logical law" is realized in the exact concept of a "generally valid formula" of this calculus, which, in turn, makes the concept of "logical law » relative. However, the boundaries of this relativity are also assumed to be the type of logical calculus. At the same time, the type of calculus, as a rule, is not a matter of arbitrary choice, but is dictated (or prompted) by the “logic of things” that they want to talk about, as well as our subjective confidence in one or another character of this logic. Calculi based on the same hypothesis about the nature of the "logic of things" are equivalent in the sense that they catalog the same logical laws. For example, calculi based on the hypothesis of ambiguity, despite all their external diversity, describe the same area of classical laws of logic - the world of identical truths (or tautologies), which have long received the philosophical characteristic of "eternal truths" or "truths in all possible worlds" (cm. ). The logic of things, the reflection of which historically was the logical laws of the so-called intuitionistic logic, is the logic of mental mathematical constructions - the “logic of knowledge”, and not the “logic of being”.

Logical laws are different from logical rules of inference. The former represent a class of generally valid expressions and are formulated in the object calculus language. The second serve to describe the facts logical consequence(see ) some expressions from others, not necessarily generally valid, and are formulated in the metalanguage of calculus. Unlike the laws of logic, the rules of inference have the form of prescriptions and are, in essence, of a normative nature. When constructing calculi, it is impossible to do without the rules of inference, but, in principle, it is possible to do without the laws of logic (this is exactly what is done in the calculus of natural inference). Nevertheless, the study of logical laws forms a natural starting point for the logical analysis of acceptable (logically correct) ways of reasoning (inference), since the concept of "acceptable" or "logically correct" reasoning is specified through the concept of "logical law".

In traditional formal logic, the term "law of logic" had a narrow meaning and applied only to the four so-called fundamental laws of correct thinking - the law of identity, the law of non-contradiction, the law of the excluded middle, and the law of sufficient reason:

The law of identity. In the process of inference, every statement and judgment must remain identical to itself (see).
The law of non-contradiction. Two opposing propositions cannot be true at the same time and in the same respect (see).
Law of the excluded middle. Of the two contradictory judgments, one is true, the other is false, and the third is not given (see).
Law of sufficient reason. No judgment can be asserted without sufficient reason (see).

This "canonization" of the term "law of logic" is currently a tribute to tradition and does not correspond to the actual state of affairs. Nevertheless, these laws can be accepted in the methodological sense as certain principles (or postulates) of theoretical thinking, since they are the most general and are used when operating with concepts and judgments, in conclusions, proofs and refutations, and therefore are present in almost all logical systems. .

In this sense identity law(lex identitatis) is interpreted as the principle of constancy or the principle of preservation of the subject and semantic meanings of judgments (statements) in some known or implied context (in conclusion, proof, theory). In the language of logical calculus, this preservation is usually expressed by the formula A ⊃ A. Adoption of the Law of Identity for Judgment A does not mean, generally speaking, acceptance of the A. But if A accepted, then the law of identity is accepted with necessity for calculi with a valid formula A ⊃ (A ⊃ A). For calculi involving negation, this reduction of the abstraction of the constancy of a proposition to the acceptance of the proposition itself has the form of a law: A ⊃ ¬ ( A ⊃ A) ⊃ ¬ A), i.e., if the law of identity is denied when a judgment is admitted, then this judgment itself is thus denied.

Law of non-contradiction(lex contradictionis) indicates the inadmissibility of the simultaneous assertion (in reasoning, in a text or theory) of two judgments, of which one is a logical negation of the other, that is, judgments of the form A and ¬ A or their conjunctions, or equivalences, or - in a broader sense - statements about the identity of obviously different objects, since usually the rules of logic are such that they allow arbitrary judgments to be derived from a contradiction, which devalues the meaningful meaning of inferences or theories. The presence of a contradiction in reasoning (theory) creates a paradoxical situation and often indicates the incompatibility of the premises underlying the reasoning (theory). This circumstance is often used in circumstantial evidence.

Law of the excluded middle(lex exclusii tertii) in logical language is written by the formula A ⌵ ¬ A and claims that there is nothing in between (intermediate assessment) between the members of an inconsistent pair (hence another Latin name for this law - tertium non datur). In methodological terms, this law expresses a constructively unjustified idea of the decidability (potentially feasible indication of truth or falsity) of an arbitrary judgment. Unlike the formula corresponding to the law of contradiction, the formula corresponding to the law of the excluded middle is not derivable in intuitionistic and constructive calculi, although it is irrefutable in them. The dichotomy of established truth and falsehood is undeniable, but the dichotomy of affirmation and negation has been disputed more than once. The most consistent criticism of the law of the excluded middle was given by L. E. Ya. Brouwer. In the light of his critique, this law should only be regarded as a principle (or postulate) of classical logic.

Law of Sufficient Reason(lex rationis determinatis seu sufficientis) expresses the methodological requirement of the validity of any knowledge, any judgment that we would like to take as a reflection of the true (real) state of things. In this sense, it is applicable not only to inferential knowledge (in particular, to the axioms and postulates of scientific theories), but also to the entire field of factual truths that are not related to formal logic. It is no coincidence that G. W. Leibniz, who introduced this principle into scientific use, attributed it primarily not to logic, but to all events that happen in the world.

In the applications of logical laws to specific situations, their common feature is revealed with particular clarity: they are all tautologies and do not carry meaningful, "substantive" information. This - general schemes, distinguishing feature which is that, substituting in them any specific statements (both true and false), we will definitely get a true expression. The indicated laws of thought have the same meaning in logic as axioms (see) or postulates have in mathematics and have the same formal character as the formulas of algebra: the latter do not say in relation to which numerical values they are fulfilled, but the laws of thought do not contain substantive characteristics, that is, they do not qualify what exactly should or should not be identified, what exactly and what should or should not contradict, and so on. This is precisely their generalizing character as operational directives of correct thinking and reasoning.

Among the many logical laws, logic distinguishes four basic, expressing the fundamental properties of logical thinking - its certainty, consistency, consistency and validity. These are the laws identity, non-contradiction, excluded third and sufficient reason. They act in any reasoning, in whatever logical form it proceeds and whatever logical operation it performs. Along with the basic ones, logic studies the laws of double negation, contra-position, de Morgan and many others, which also operate in thinking, causing the correct connection of thoughts in the process of reasoning.

Consider the basic logical laws.

The law of identity. Any thought in the process of reasoning must have a definite, stable content. This fundamental property of thinking - its certainty - expresses the law of identity: every thought in the process of reasoning must be identical to itself(and there is a, or a=a, where under A any thought is understood).

The law of identity can be expressed by the formula R R(If R, That R), Where R- any statement, - a sign of implication.

It follows from the law of identity: one cannot identify different thoughts, one cannot take identical thoughts for non-identical ones. Violation of this requirement in the process of reasoning is often associated with a different expression of the same thought in the language.

For example, two judgments: “N. committed theft" and "N. secretly stole someone else's property "- they express the same thought (if, of course, we are talking about the same person). The predicates of these judgments are equivalent concepts: theft is the secret theft of someone else's property. Therefore, it would be erroneous to consider these thoughts as non-identical.

On the other hand, the use of ambiguous words can lead to erroneous identification of different thoughts. For example, in criminal law, the word "fine" denotes a measure of punishment provided for by the Criminal Code, in civil law This word denotes a measure of administrative influence. Obviously, such a word should not be used in one sense.

The identification of different thoughts is often associated with differences in profession, education, etc. This happens in investigative practice, when the accused or the witness, not knowing the exact meaning of certain concepts, understands them differently than the investigator. This often leads to confusion, ambiguity, and makes it difficult to clarify the essence of the case.

The identification of different concepts is a logical fallacy - change of concept which can be either unconscious or intentional.

Compliance with the requirements of the law of identity is important in the work of a lawyer, which requires the use of concepts in their exact meaning.

In any case, it is important to find out the exact meaning of the terms used by the accused or the witnesses, and to use these terms in a strictly defined sense. Otherwise, the subject of thought will be missed and instead of clarifying the matter, it will be confused.

The law of non-contradiction. Logical thinking is characterized by consistency. Contradictions destroy thought, complicate the process of cognition. The requirement of consistency of thinking expresses the formal-logical law of non-contradiction: two judgments incompatible with each other cannot be true at the same time; at least one of them must be false .

This law is formulated as follows: it is not true that a and not-a (two thoughts cannot be true, one of which denies the other). It is expressed by the formula ù (RÙù R) (it is not true that p and not-p are both true). Under R means any statement ù r- negation of the statement R, sign ù before the whole formula - the negation of two statements connected by a conjunction sign.

The law of non-contradiction applies to all incompatible judgments.

To understand it correctly, the following must be kept in mind. In asserting something about any object, one cannot, without contradicting oneself, deny (1) the same thing (2) about the same object, (3) taken at the same time and (4) in the same respect .

It is clear that there will be no contradiction between judgments if one of them asserts belonging to the subject one sign, and in the other, belonging to the same subject is denied other sign (1) and if talking about different subjects (2).

There will be no contradiction even if we affirm something and deny the same thing about one person, but considered V different time. Let us assume that the accused N. at the beginning of the investigation gave false evidence, but at the end of the investigation he was forced under the weight of the evidence incriminating him to confess and give true testimony. In this case, the judgments: “The testimonies of the accused N. are false” and “The testimonies of the accused N. are true” do not contradict each other.

(4) Finally, the same object of our thought can be considered in different relationships. So, about student Shchukin, one can say that he knows German well, since his knowledge satisfies the requirements for entering the institute. However, this knowledge is not enough to work as a translator. In this case, we have the right to say: "Shchukin does not know German well." In two judgments, knowledge by Shchukin German language considered from the point of view of different requirements, therefore, these judgments also do not contradict each other.

The law of non-contradiction expresses one of the fundamental properties of logical thinking - consistency, consistency of thinking. Its conscious use helps to detect and eliminate contradictions in one's own and other people's reasoning, develops a critical attitude towards all kinds of inaccuracies, inconsistencies in thoughts and actions.

N.G. Chernyshevsky emphasized that inconsistency in thoughts leads to inconsistency in actions. Whoever does not understand the principles in all logical completeness and sequence, he wrote, has not only confusion in his head, but also nonsense in his affairs.

The ability to reveal and eliminate logical contradictions, often found in the testimony of witnesses, the accused, the victim, plays an important role in judicial and investigative practice.

One of the main requirements for a version in a forensic study is that, when analyzing the totality of factual data on the basis of which it is built, these data do not contradict each other and the version put forward as a whole. The presence of such contradictions should attract the most serious attention of the investigator. However, there are cases when the investigator, having put forward a version that he considers plausible, does not take into account the facts that contradict this version, ignores them, and continues to develop his version in spite of the contradictory facts.

During the trial, the prosecutor and the defender, the plaintiff and the defendant put forward positions that contradict each other, defending their arguments and challenging the arguments of the opposite side.

Therefore, it is necessary to carefully analyze all the circumstances of the case so that the final decision of the court is based on reliable and consistent facts.

Contradictions in judicial acts are inadmissible. Among the circumstances under which the verdict is recognized as inappropriate to the actual circumstances of the case, criminal procedural law includes significant contradictions contained in the conclusions of the court set out in the verdict.

Law of the excluded middle. The law of non-contradiction applies to all incompatible judgments. It establishes that one of them must be false. The question of the second proposition remains open: it may be true, but it may also be false.

The law of the excluded middle applies only to contradictory (contradictor) judgments. It is formulated as follows: two contradictory propositions cannot be false at the same time, one of them must be true: A is either b, or non-b. Either the statement of a fact is true, or its negation.

contradictory(contradictor) are judgments, in one of which something is affirmed (or denied) about everyone subject of a certain set, and in another - is denied (asserted) about some part this set. These judgments cannot be both true and false: if one of them is true, then the other is false, and vice versa. For example, if the judgment "To every citizen Russian Federation the right to receive qualified legal assistance is guaranteed” is true, then the proposition “Some citizens of the Russian Federation are not guaranteed the right to receive qualified legal assistance” is false. There are also two contradictory statements about one object, in one of which something is affirmed, and in the other the same thing is denied. For example: "P. brought to administrative responsibility" and "P. not held administratively liable." One of these judgments is necessarily true, the other is necessarily false.

This law can be written using the disjunction: r v ur, Where R- any statement ù r- negation of the statement R.

Like the law of non-contradiction, the law of the excluded middle expresses the consistency, consistency of thinking, does not allow contradictions in thoughts. At the same time, acting only in relation to contradictory judgments, he establishes that two contradictory judgments cannot be not only simultaneously true (as indicated by the law of non-contradiction), but also simultaneously false: if one of them is false, then the other must be true, There is no third.

Of course, the law of the excluded middle cannot indicate which of these judgments is true. This issue is resolved by other means. The significance of the law lies in the fact that it indicates the direction in the search for truth: only two solutions to the problem are possible, and one of them (and only one) is necessarily true.

The law of the excluded middle requires clear, definite answers, pointing to the impossibility of answering the same question in the same sense both “yes” and “no”, to the impossibility of looking for something in between affirming something and denying the same.

This law is of great importance in legal practice, where a categorical solution of the issue is required. The lawyer must decide the case in the form of "either-or". This fact is either established or not established. The accused is either guilty or not guilty. Jus (right) knows only: "either-or".

Law of sufficient reason. Our thoughts about any fact, phenomenon, event can be true or false. Expressing a true thought, we must substantiate its truth, i.e. prove its validity. So, when bringing charges against the defendant, the accuser must provide the necessary evidence, substantiate the truth of his assertion. Otherwise, the accusation will be unfounded.

The requirement of proof, the validity of thought expresses the law of sufficient reason: any thought is recognized as true if it has a sufficient basis. If there b, that is, its base A.

A person's personal experience can be a sufficient basis for thoughts. The truth of some judgments is confirmed by their direct comparison with the facts of reality. So, for a person who witnessed a crime, the justification for the truth of the judgment “N. committed a crime” will be the very fact of the crime of which he was an eyewitness. But personal experience is limited. Therefore, a person in his activities has to rely on the experience of other people, for example, on the testimony of eyewitnesses of an event. Such grounds are usually resorted to in investigative and judicial practice when investigating crimes.

Thanks to the development of scientific knowledge, a person is increasingly using the experience of all mankind as the basis of his thoughts, enshrined in the laws and axioms of science, in the principles and provisions that exist in any field of human activity.

The truth of laws, axioms has been confirmed by the practice of mankind and therefore does not need new confirmation. To confirm any particular case, there is no need to substantiate it with the help of personal experience. If, for example, we know the law of Archimedes (each body immersed in a liquid loses as much in its weight as the liquid displaced by it weighs), then there is no point in immersing an object in a liquid in order to find out how much it loses in weight. The law of Archimedes will be a sufficient basis for confirming any particular case.

Thanks to science, which in its laws and principles consolidates the socio-historical practice of mankind, in order to substantiate our thoughts, we do not each time resort to checking them, but justify them logically, by deriving from already established provisions.

Thus, any other, already tested and established thought, from which the truth of this thought necessarily follows, can be a sufficient basis for any thought.

If from the truth of judgment A follows the truth of the proposition b, That A will be the basis for b, a b- a consequence of this foundation.

The connection between the foundation and the effect is a reflection in thinking of objective, including cause-and-effect relationships, which are expressed in the fact that one phenomenon (cause) gives rise to another phenomenon (effect). However, this reflection is not direct. In some cases, the logical basis may coincide with the cause of the phenomenon (if, for example, the idea that the number of traffic accidents has increased is justified by pointing to the cause of this phenomenon - ice on the roads). But most often there is no such coincidence. The judgment "It has rained recently" can be substantiated by the judgment "The roofs of the houses are wet"; the trace of the protectors of automobile awls is a sufficient basis for the judgment "B this place a car has passed." Meanwhile, wet roofs and a trail left by a car are not the cause, but the consequence of these phenomena. Therefore, the logical connection between the basis and the effect must be distinguished from the causal relationship.

Validity is the most important property logical thinking. In all cases when we affirm something, convince others of something, we must prove our judgments, give sufficient reasons confirming the truth of our thoughts. This is the fundamental difference scientific thinking from unscientific thinking, which is characterized by lack of evidence, the ability to take various positions and dogmas on faith. This is especially characteristic of religious thinking, which is based not on proof, but on faith.

The law of sufficient reason is incompatible with various prejudices and superstitions. For example, there are absurd signs: to break a mirror - unfortunately, to sprinkle salt - to a quarrel, etc., although there is no causal relationship between a broken mirror and misfortune, spilled salt and a quarrel. Logic is the enemy of superstition and prejudice. It requires the validity of judgments and is therefore incompatible with statements that are built according to the scheme "after this - therefore, because of this." This logical fallacy occurs when causality is confused with a simple sequence in time, when the antecedent phenomenon is taken as the cause of the subsequent one.

The law of sufficient reason is of great theoretical and practical importance. Fixing attention on the judgments that justify the truth of the put forward provisions, this law helps to separate the true from the false and come to the right conclusion.

The significance of the law of sufficient reason in legal practice is, in particular, as follows. Any conclusion of the court or investigation must be substantiated. In the materials concerning any case, containing, for example, the allegation of the guilt of the accused, there must be data that is a sufficient basis for the accusation. Otherwise, the accusation cannot be recognized as correct. The issuance of a reasoned sentence or court decision in all, without exception, cases is the most important principle of procedural law.

The language of logic

The necessary connection between thinking and language, in which language acts as a material shell of thoughts, means that the identification of logical structures is possible only through the analysis of linguistic expressions. Just as the kernel of a nut can be reached only by opening its shell, so logical forms can only be revealed by analyzing the language.

In order to master logical-linguistic analysis, let us briefly consider the structure and functions of the language, the relationship between logical and grammatical categories, as well as the principles for constructing a special language of logic.

Language is a sign information system that performs the function of forming, storing and transmitting information in the process of cognition of reality and communication between people.

The main building material in the construction of the language are the signs used in it. Sign - it is any sensually perceived (visually, aurally or otherwise) object that acts as a representative of another object. Among the various signs, we distinguish two types: signs-images and signs-symbols.

Signs-images have a certain resemblance to the designated objects. Examples of such signs: copies of documents; fingerprints; photographs; some road signs depicting children, pedestrians and other objects. Signs-symbols have no resemblance to the designated objects. For example: musical signs; Morse code characters; letters in the alphabets of national languages.

The set of initial signs of the language makes it alphabet.

A comprehensive study of the language is carried out by the general theory of sign systems - semiotics, which analyzes the language in three aspects: syntactic, semantic and pragmatic.

Syntax- This is a section of semiotics that studies the structure of a language: the ways of formation, transformation and connections between signs. Semantics deals with the problem of interpretation, i.e. analysis of the relationship between signs and designated objects. pragmatics analyzes the communicative function of the language - emotional, psychological, aesthetic, economic and other relations of a native speaker to the language itself.

By origin, languages are natural and artificial,

natural languages- these are sound (speech) and then graphic (writing) information sign systems that have historically developed in society. They arose to consolidate and transfer the accumulated information in the process of communication between people. Natural languages act as carriers of the centuries-old culture of peoples. They are distinguished by rich expressive possibilities and universal coverage of various areas of life.

Constructed languages- these are auxiliary sign systems created on the basis of natural languages for accurate and economical transmission of scientific and other information. They are constructed using natural language or a previously constructed artificial language. A language that acts as a means of building or learning another language is called metalanguage, basic - language-object. The metalanguage, as a rule, has richer expressive possibilities compared to the object language.

Artificial languages of varying degrees of severity are widely used in modern science and technology: chemistry, mathematics, theoretical physics, computer technology, cybernetics, communications, shorthand.

A special group is mixed languages, the base in which is the natural (national) language, supplemented by symbols and conventions related to a specific subject area. This group includes the language conventionally called "legal language" or "language of law". It is built on the basis of a natural (in our case, Russian) language, and also includes many legal concepts and definitions, legal presumptions and assumptions, rules of proof and refutation. The initial cell of this language is the norms of law, combined into complex legal systems.

Artificial languages are also successfully used by logic for precise theoretical and practical analysis of mental structures.

One of these languages is the language of propositional logic. It is applied in a logical system called calculus of propositions, which analyzes reasoning based on the truth characteristics of logical connectives and abstracting from the internal structure of judgments. The principles for constructing this language will be outlined in the chapter on deductive reasoning.

The second language is the language of predicate logic. He is used in a logical system called predicate calculus, which, when analyzing reasoning, takes into account not only the truth characteristics of logical connectives, but also the internal structure of judgments. Let us briefly consider the composition and structure of this language, the individual elements of which will be used in the course of a meaningful presentation of the course.

Intended for the logical analysis of reasoning, the language of predicate logic structurally reflects and closely follows the semantic characteristics of natural language. The main semantic (semantic) category of the language of predicate logic is the concept of a name.

Name - This is a linguistic expression that has a specific meaning in the form of a single word or phrase, denoting or naming some extralinguistic object. The name as a linguistic category thus has two obligatory characteristics or meanings: subject meaning and semantic meaning.

The subject meaning (denotation) of a name is one or a set of any objects that are denoted by this name. For example, the denotation of the name "house" in Russian will be the whole variety of structures that this name denotes: wooden, brick, stone; single-storey and multi-storey, etc.

The semantic meaning (meaning, or concept) of a name is information about objects, i.e. their inherent properties, with the help of which a variety of objects are distinguished. In the example above, the meaning of the word "house" will be following characteristics any house: 1) this structure (building), 2) built by man, 3) intended for habitation.

The relationship between name, meaning and denotation (object) can be represented by the following semantic scheme:

This means that the name denotates, i.e. designates objects only through meaning, and not directly. A linguistic expression that has no meaning cannot be a name, since it is not meaningful, and therefore not objectified, i.e. has no denotation.

The types of names of the predicate logic language, determined by the specifics of the naming objects and representing its main semantic categories, are the names of: 1) objects, 2) attributes and 3) sentences.

Item names denote single objects, phenomena, events or their sets. The object of research in this case can be both material (airplane, lightning, pine) and ideal (will, legal capacity, dream) objects.

The composition distinguishes names simple, which do not include other names (state), and complex, including other names (Earth satellite). By denotation, names are single And are common. A single name denotes one object and is represented in the language by a proper name (Aristotle) or given descriptively (the largest river in Europe). The common name denotes a set consisting of more than one object; in the language it can be represented by a common name (law) or given descriptively (big wooden house).

Feature names- qualities, properties or relationships are called predictors. In a sentence, they usually play the role of a predicate (for example, "be blue", "run", "give", "love", etc.). The number of item names to which the predictor refers is called its terrain. Predicators expressing the properties inherent in individual objects are called single (e.g. "the sky is blue"). Predicators that express relationships between two or more things are called multi-seat. For example, the predicator "to love" refers to two-places ("Mary loves Peter"), and the predicator "to give" - to three-places ("Father gives a book to his son").

Offers- they are names for language expressions in which something is affirmed or denied. According to their logical meaning, they express true or false.

Predicate Logic Language Alphabet includes the following types of signs (symbols):

1) a, b, c,... - symbols for single (proper or descriptive) names of objects; they are called subject constants, or constants;

2) x, y, z,... - symbols of common names of objects that take values in one or another area; they are called subject variables;

3) R 1 , Q 1 , R 1,... - symbols for predicates, indices over which express their locality; they are called predicate variables;

4) p, q, r,... - symbols for statements, which are called propositional, or propositional variables (from the Latin propositio - "statement");

5) ",$ - symbols for quantitative characteristics of statements; they are called quantifiers: "- general quantifier; it symbolizes expressions - everything, everyone, everyone, always, etc.; $ - existential quantifier; it symbolizes expressions - some, sometimes, happens, occurs, exists, etc.;

6) logical links:

Ù - conjunction (conjunction "and");

v - disjunction (conjunction "or");

® - implication (conjunction "if..., then...");

º - equivalence, or double implication (conjunction "if and only if..., then...");

ù - negation ("it is not true that...").

Technical characters of the language: (,) - left and right brackets.

This alphabet does not include other characters. Permissible, i.e. expressions that make sense in the language of predicate logic are called well-formed formulas - PPF. concept PPF is introduced by the following definitions:

1. Any propositional variable - p, q, r,... There is PPF.

2. Any predicate variable, taken with a sequence of object variables or constants, the number of which corresponds to its locality, is PPF: A 1 (x), A 2 (x, y), A 3 (x, y, z ), A n (x, y,..., n), Where A 1, A 2, A 3,..., A n- signs of metalanguage for predicators.

3. For any formula with object variables, in which any of the variables is associated with a quantifier, the expressions "xA(x) And $ xA(x) will also PPF.

4. If A And IN- formulas ( A And IN- signs of the metalanguage for expressing schemes of formulas), then the expressions:

A u B,

A ® B,

A º B,

u A, u B

are also formulas.

5. Any other expressions, in addition to those provided for in paragraphs 1-4, are not PPF given language.

With the help of the given logical language, a formalized logical system is built, called the predicate calculus. Elements of the language of predicate logic will be used in what follows to analyze individual fragments of natural language.

§ 5. History of Logic (Short Outline)

As an independent science, logic developed more than two thousand years ago, in the 4th century. BC. Its founder is the ancient Greek philosopher Aristotle(348-322 BC). In his logical works, which received the general name "Organon" (Greek "instruments of knowledge"), Aristotle formulated the basic laws of thinking: identities, contradictions and the excluded middle, described the most important logical operations, developed a theory of concepts and judgments, studied in detail the deductive (syllogistic) inference. The Aristotelian doctrine of syllogism formed the basis of one of the areas of modern mathematical logic - the logic of predicates. An important stage in the development of the teachings of Aristotle was the logic ancient stoics(Zeno, Chrysippus, etc.), who supplemented the Aristotelian theory of syllogism with a description of complex inferences. The logic of the Stoics is the basis of another direction of mathematical logic - the logic of propositions.

Among other ancient thinkers who developed and commented on the logical teachings of Aristotle, one should mention Galena whose name is given to the 4th figure of the categorical syllogism; porfiria, known for his visual diagram, which displays the relationship of subordination between concepts ("Porfiry's tree"); Boethia, whose writings long time served as the main logical aids.

Logic also developed in the Middle Ages, but scholasticism distorted the teachings of Aristotle, adapting it to justify religious dogma.

Significant progress in logical science in modern times. The most important stage in its development was the theory of induction developed by the English philosopher F. Bacon(1561-1626). Bacon criticized the deductive logic of Aristotle, distorted by medieval scholasticism, which, in his opinion, cannot serve as a method scientific discoveries. This method should be induction, the principles of which are set forth in his work The New Organon (in contrast to the old, Aristotelian Organon). The development of the inductive method is a great merit of Bacon, but he unjustly opposed it to the method of deduction; In fact, these methods do not exclude, but complement each other. Bacon developed methods of scientific induction, subsequently systematized by the English philosopher and logician J.S. Millem(1806-1873).

The deductive logic of Aristotle and the inductive logic of Bacon - Mill formed the basis of a general educational discipline, which for a long time was an indispensable element of the European education system and forms the basis of logical education at the present time.

This logic is called formal, since it arose and developed as a science of the forms of thinking. It is also called traditional or Aristotelian logic.

The further development of logic is associated with the names of such prominent Western European thinkers as R. Descartes, G. Leibniz, I. Kant and others.

French philosopher R. Descartes(1569-1650) criticized medieval scholasticism, he developed the ideas of deductive logic, formulated the rules of scientific research, set forth in the essay "Rules for the Guidance of the Mind". In 1662, the book “Logic, or the Art of Thinking” was published in Paris, written by the followers of Descartes A. Arnaud and P. Nicol, also known as the Logic of Port-Royal. The book had a noticeable influence on the entire subsequent history of the development of logic.

A major contribution to the study of logical problems was made by the German philosopher G. Leibniz(1646-1716), who formulated the law of sufficient reason, put forward the idea of mathematical logic, which was developed only in the 19th-20th centuries; German philosopher I. Kant(1724-1804) and many other Western European philosophers and scientists.

Significant achievements in the development of the logic of Russian philosophers and scientists. A number of original ideas were put forward M.V. Lomonosov(1711- 1765), A.N. Radishchev(1749-1802), N.G. Chernyshevsky(1828-1889). Known for their innovative ideas in the theory of inference, Russian logicians M.I. carian(1804-1917) and L.V. Rutkovsky(1859-1920). One of the first to develop the logic of relations between the philosopher and the logician S.I. Povarnin(1807-1952).

In the second half of the XIX century. in logic, the methods of calculus developed in mathematics are beginning to be widely used. This direction is developed in the works of D. Boole, W.S. Jevons, P.S. Poretsky, G. Frege, C. Pierce, B. Russell, J. Lukasiewicz and other mathematicians and logicians. The theoretical analysis of deductive reasoning by methods of calculus using formalized languages is called mathematical, or symbolic, logic.

Symbolic logic- an intensively developing field of logical research, including many branches, or, as they are commonly called, “logics” (for example, propositional logic, predicate logic, probabilistic logic, etc.). Much attention is paid to the development multivalued logic, in which, in addition to the two truth values accepted in traditional logic - "true" and "false" - many truth values are allowed. So, in the developed by the Polish logician J. Lukasiewicz (1878-1956) three-valued logic a third meaning is introduced - "possibly" ("neutral"). They built modal logic system co the meanings of “possible”, “impossible”, “necessary”, etc., as well as four-digit And infinite value logic.

The most promising sections are probabilistic logic, examining statements that take many degrees of likelihood - from 0 to 1, timing logic and many others.

Of particular importance for jurisprudence is the section of modal logic called deontic logic, investigating the structures of the language of prescriptions, i.e. statements with the meaning "mandatory", "permitted", "forbidden", "indifferent", which are widely used in law-making and law enforcement activities.

The study of reasoning processes in systems of symbolic logic had a significant impact on the further development of formal logic as a whole. At the same time, symbolic logic does not cover all the problems of traditional formal logic and cannot completely replace the latter. These are two directions, two stages in the development of formal logic.

The peculiarity of formal logic is that it considers the forms of thinking, abstracting from their emergence, change, development. This side of thinking is studied dialectical logic, for the first time in an expanded form presented in an objective-idealistic philosophical system Hegel(1770-1831) and revised from a materialist standpoint in the philosophy of Marxism.

Dialectical logic studies the laws of development of human thinking, as well as the principles and requirements that are formed on their basis. These include the objectivity and comprehensiveness of the consideration of the subject, the principle of historicism, the bifurcation of the one into opposite sides, the ascent from the abstract to the concrete, the principle of the unity of the historical and the logical, etc. Dialectical logic serves as a method of cognizing the dialectics of the objective world.

Formal logic and dialectical logic study the same object - human thinking, but each of them has its own subject of study. This means that dialectical logic does not and cannot replace formal logic. These are two sciences of thinking, they develop in close interaction, which is clearly manifested in the practice of scientific and theoretical thinking, which uses both the formal logical apparatus and the means developed by dialectical logic in the process of cognition.

Formal logic studies the forms of thinking, revealing the structure common to thoughts that are different in content. Considering, for example, a concept, it studies not the specific content of various concepts (this is the task of the special sciences), but the concept as a form of thinking, regardless of what kind of objects are conceived in concepts. When studying a judgment, logic abstracts from their specific content, revealing a structure common to judgments that are different in content. Formal logic studies the laws that determine the logical correctness of thinking, without which it is impossible to come to results that correspond to reality, to know the truth.

Thinking that does not obey the requirements of formal logic is not capable of correctly reflecting reality. Therefore, the study of thinking, its laws and forms must begin with formal logic, the presentation of the foundations of which is the task of the proposed textbook.

Logic value

Human thinking is subject to logical laws and proceeds in logical forms, regardless of the science of logic. People think logically without knowing its rules, just as they speak correctly without knowing the rules of grammar.

But does it follow from this that the study of logic has no practical value?

Proponents of this view sometimes refer to Hegel's ironic remark that logic "teaches" to think, just as physiology "teaches" to digest. Of course, one can think correctly without studying logic, speak correctly without knowing grammar, digest food without knowing physiology. However, the practical significance of these sciences cannot be underestimated. When Academician I.P. Pavlov was asked what he sees as the main goals of physiological science, the great Russian physiologist answered: “The task of physiology is to teach a person how to eat, breathe, how to work and rest correctly in order to live as long as possible.”

As for logic, its task is to teach a person to consciously apply the laws and forms of thinking and, on the basis of this, think more logically and, therefore, more correctly cognize the world around.

Knowledge of logic increases the culture of thinking, develops the ability to think more "competently", develops a critical attitude towards one's own and other people's thoughts. Therefore, the opinion that the study of logic has no practical value is untenable.

Many great philosophers, outstanding figures of science and culture: Plato and Hobbes, Lomonosov and Chernyshevsky, Timiryazev and Ushinsky - attached great importance to the study of logic, knowledge of its laws, pointed out the need to develop the ability for logical thinking. “No matter how one relates to the question of whether our ability to find correct arguments increases as a result of studying logic or not,” says the famous American logician and mathematician S. Kleene, “it is indisputable that as a result of studying logic, the ability to check the correctness of reasoning increases. After all, logic provides methods for analyzing reasoning ... Even if we believe that we ourselves can not make mistakes in our reasoning, we still have no doubt that there are many who are inclined to make mistakes (especially among those who disagree with us).

"Logic - essential tool, freeing from unnecessary, unnecessary memorization, helping to find in the mass of information that valuable thing that a person needs, - wrote the famous physiologist academician N.K. Anokhin. “Any specialist needs it, whether he is a mathematician, a physician, a biologist.”

To think logically means to think accurately and consistently, not to allow contradictions in one's reasoning, to be able to reveal logical errors. These qualities of thinking are of great importance in any field of scientific and practical activity, including in the work of a lawyer, which requires accuracy of thinking, validity of conclusions.

The best Russian lawyers were distinguished not only by their deep knowledge of all the circumstances of the case and the brightness of their speeches, but also by their strict logic in the presentation and analysis of the material, and the irrefutable argumentation of the conclusions. Here, for example, is how the professional skills of the famous Russian lawyer of the second half of the last century, P.A. Alexandrova: “The most characteristic for the judicial oratorical skills of P.A. Alexandrov is the firm logic and consistency of his judgments, the ability to carefully weigh and determine the place of any evidence in the case, as well as to convincingly argue and substantiate his most important arguments. A.F. Koni emphasized the "irresistible logic" in the speeches of V.D. Spasovich. Strict consistency, logic and persuasiveness are noted in the speeches of the prominent lawyer K.F. Khaltulari.

Conversely, inconsistent and contradictory reasoning makes it difficult to identify the case, and in some cases may be the cause of miscarriage of justice.

Knowledge of logic helps a lawyer prepare a logically coherent, well-argued speech, reveal contradictions in the testimony of the victim, witnesses, the accused, refute the unfounded arguments of his opponents, build a judicial version, outline a logically consistent plan for examining the scene, draw up an official document consistently, consistently and reasonably, etc. .d. All this has importance in the work of a lawyer aimed at strengthening the rule of law and the rule of law.

CONTROL QUESTIONS

1. What is sensory knowledge, in what forms does it take place?

2. What is thinking, what is its role in cognition?

3. What is a form of thinking?

4. What is the difference between the truth of thought and the logical correctness of reasoning?

Of the four laws of thinking in traditional logic, Aristotle established at least two - the laws (prohibitions) of contradiction and the excluded third. . The laws of identity and sufficient reason for Aristotle are also outlined in the doctrine of scientific knowledge as evidence-based knowledge (the law of sufficient reason) and in the thesis according to which “it is impossible to think anything if you do not think [every time] one thing” (Aristotle. Metaphysics , IV, 4, p. 64) - the law of identity.

Law [prohibition] of contradiction in a short form it sounds like “it is impossible to exist and not exist together” (ibid., p. 63) or: “The same thing cannot be and not be at the same time” (XI, 5, p. 187), and in full - as a statement: "It is impossible that the same thing together (jointly, simultaneously) be and not be inherent in the same thing in the same sense" (IV, 3, p. 63). In "Metaphysics" by Aristotle, the logical aspect of the law [prohibition] of contradiction is also formulated in the words that "it is impossible to speak correctly, simultaneously affirming and denying something" (IV, 6, p. 75). This aspect is more clearly shown in the logical works of Aristotle, where it is repeatedly stated that it is impossible to affirm and deny the same thing at the same time. This law cannot be directly substantiated, but it is possible to refute the opposite view by showing its absurdity. Anyone who disputes the law [prohibition] of contradiction uses it. Further, if this logical law is not recognized, everything will become an indistinguishable unity. This also includes the considerations of Aristotle against the skeptic, who, arguing that everything is true or that everything is false, which turns out to be absurd from the point of view of practice, can do this only by rejecting the law [of prohibition] of contradiction.

Aristotle. Sculpture by Lysippos

Speaking about this basic law of logical thinking, Aristotle takes into account the extremes that the researchers who approached his discovery fell into. For example, the Cynic Antisthenes believed that it was necessary to say “man is a man”, but one cannot say that “man is a living being” or “white” or “educated”, because this would mean some kind of “violation”. In the light of the law discovered by Aristotle, Antisthenes can be better understood. Arguing that "a person is educated", we argue that "a is not-a", because "educated" is not the same as "man". It would seem that the law [forbidding] contradiction confirms this. It turns out that the statement "a person is educated" means that a person is both a [man] and not-a [educated].

Aristotle objects: there is no a and no-a here, a person is opposed not by an “educated”, but by a non-person, because the contradiction can only be within the same category, and “man” and “educated” refer to different categories(“man” is the essence, and “educated” is the quality).

The logical law [of prohibition] of contradiction has raised many objections. Hegel criticized Aristotle, arguing that this law actually forbids formation, change, development, that it is metaphysical. But the objection testifies to Hegel's misunderstanding of the essence of this law. In the logic of Aristotle, the law [of prohibition] of contradictions is absolute, but it operates only in the sphere of actual being, and it does not operate in the sphere of the possible. Therefore, formation, according to Aristotle, exists as the realization of one of the possibilities, which, being realized, actualized, excludes other possibilities, but only in reality, and not in possibility. If an updated possibility becomes just a possibility again, it will be replaced by another updated possibility. Having defined the boundaries of his formal logic, Aristotle thereby left room for dialectical logic. Potentially existent is dialectical, actual existent is relatively non-dialectical.

In the logic of Aristotle, one can find other fundamental restrictions on the scope of the law of contradiction. Its action does not extend to the future, but it is nevertheless connected with the same sphere of possibility, since the future is fraught with many possibilities, while the present is poor, since only one thing is actualized, but it is potentially rich. The past, on the other hand, is poor in its actuality, excluding potentiality, because in the past there are no longer any possibilities, except for the realized, the occurred, and not subject to change.

The aggravated form of the law [prohibition] of contradiction is the logical law of the excluded middle , which forbids not only that in relation to the same thing "b" and "not-b" cannot be true at the same time, but also that, moreover, the truth of "b" means the falsity of "not-b", and vice versa. This law in Aristotle’s Metaphysics is expressed as follows: “There can be nothing in the middle between two contradictory [each other] judgments, but about one [subject] every single predicate must either be affirmed or denied” (IV, 7, p. 75) . In Aristotle's "Second Analytics" it is said that "about anything, either affirmation or negation is true" (I, 1, p. 257).

The operation of these laws of Aristotle's logic is such that the law of contradiction does not necessarily entail the law of the excluded middle, but the law of the excluded middle implies the operation of the law of contradiction. Therefore, it was said above that the law of the excluded middle is a sharper form of the law of contradiction.

This difference in the scope of these laws of Aristotelian logic means that there are different types contradictions. Above, the contradiction proper and its softened form, the opposite, were distinguished. Both are two kinds of opposites. Later, they began to call this contradictory and contradictory contradictions. Only a contradictory contradiction is connected with both laws. An example of a contradictory opposite: "This paper is white" and "This paper is not white." There is no middle ground here. Contrary opposition is bound only by the law of the prohibition of contradiction. Example: "This paper is white" and "This paper is black", because the paper can be gray. A counter contradiction (opposite) admits a mean, a contradictory one does not. The members of a counter contradiction can be both false (when the truth is between, this is the third value), but they cannot be true at once, this is prohibited by the law of contradiction. The members of a contradictory opposition cannot be not only immediately true, but also immediately false; the falsity of one side entails the truth of the other. True, in the logic of Aristotle we do not find such accuracy.

Among the many logical laws, logic identifies four main ones that express the fundamental properties of logical thinking - its certainty, consistency, consistency and validity. These are the laws of identity, non-contradiction, excluded middle and sufficient reason. They act in any reasoning, no matter what logical form it takes and no matter what logical operation it performs. Along with the basic ones, logic studies the laws of double negation, contra-position, de Morgan and many others, which also operate in thinking, causing the correct connection of thoughts in the process of reasoning.

Consider the basic logical laws.

The law of identity. Any thought in the process of reasoning must have a definite, stable content. This fundamental property of thinking - its certainty - expresses the law of identity:

any thought in the process of reasoning must be identical to itself (a is a, or a = a, where a is understood as any thought).

The law of identity can be expressed by the formula p->p (if p, then p), where p is any statement, -> is the sign of the implication.

On the other hand, the use of ambiguous words can lead to erroneous identification of different thoughts. For example, in criminal law, the word “fine” denotes a measure of punishment provided for by the Criminal Code; in civil law, this word denotes a measure of administrative influence. Obviously, such a word should not be used in one sense.

leads to confusion, ambiguity, makes it difficult to clarify the essence of the case.

The identification of different concepts is a logical error - a substitution of a concept, which can be both unconscious and deliberate.

Compliance with the requirements of the law of identity is important in the work of a lawyer, which requires the use of concepts in their exact meaning.

In any case, it is important to find out the exact meaning of the concepts used by the accused or witnesses, and to use these concepts in a strictly defined sense. Otherwise, the subject of thought will be missed and instead of clarifying the matter, it will be confused.

The law of non-contradiction. Logical thinking is characterized by consistency. Contradictions destroy thought, complicate the process of "cognition. The requirement of consistency of thinking expresses the formal-logical law of non-contradiction: two judgments incompatible with each other cannot be true at the same time; at least one of them must be false."

This law is formulated as follows: it is not true that a and not-a (two thoughts cannot be true, one of which denies the other). It is expressed by the formula -| (p l~p) (it is not true that p and not-p are both true). Under p is understood any statement, under ~) p is the negation of the statement p, the sign ~1 in front of the whole formula is the negation of two statements connected by the conjunction sign.

The law of non-contradiction applies to all incompatible judgments2.

It is clear that there will be no contradiction between judgments if one of them affirms that the object belongs to one attribute, and the other denies that another attribute belongs to the same object (1) and if wet is about different subjects (2).

"According to tradition, this law is usually called the law of contradiction. However, the name - the law of non-contradiction - more accurately expresses its real meaning.

2 On incompatible judgments, see Ch. IV, § 4.

(3) There will be no contradiction even if we affirm something and deny the same thing about the same person, but considered at different times. Let us assume that the accused N. at the beginning of the investigation gave false evidence, but at the end of the investigation he was forced under the weight of the evidence incriminating him to confess and give true testimony. In this case, the judgments: “The testimonies of the accused N. are false” and “The testimonies of the accused N. are true” do not contradict each other.

(4) Finally, the same object of our thought can be considered in different respects. So, about student Shchukin, one can say that he knows German well, since his knowledge satisfies the requirements for entering the institute. However, this knowledge is not enough to work as a translator. In this case, we have the right to say: "Shchukin does not know German well." In two judgments, Shchukin's knowledge of the German language is considered from the point of view of different requirements, therefore, these judgments also do not contradict each other.

The law of non-contradiction expresses one of the fundamental properties of logical thinking - consistency, consistency of thinking. Its conscious use helps to detect and eliminate contradictions in one's own and other people's reasoning, develops a critical attitude to all kinds of inaccuracies, inconsistencies in thoughts and actions.

The ability to reveal and eliminate logical contradictions, often found in the testimony of witnesses, the accused, the victim, plays an important role in judicial and investigative practice.

Therefore, it is necessary to carefully analyze all the circumstances of the case so that the final decision of the court is based on reliable and consistent facts.

Law of the excluded middle. The law of non-contradiction applies to all incompatible judgments. It establishes that one of them must be false. The question of the second proposition remains open: it may be true, but it may also be false.

The law of the excluded middle applies only to contradictory (contradictor) judgments. It is formulated as follows: two contradictory propositions cannot be simultaneously false, one of them is necessarily true: a is either b or not-b. Either the statement of a fact is true, or its negation.

Contradictory (contradictory) are judgments, in one of which something is affirmed (or denied) about each object of a certain set, and in the other - something is denied (asserted) about some part of this set. These judgments cannot be both true and false: if one of them is true, then the other is false, and vice versa. For example, if the proposition “Every citizen of the Russian Federation is guaranteed the right to receive qualified legal assistance” is true, then the proposition “Some citizens of the Russian Federation are not guaranteed the right to receive qualified legal assistance” is false. Contradictory are also two judgments about one subject, in one of which something is affirmed, and in the other the same thing is denied. For example: "P. brought to administrative responsibility" and "P. not held administratively liable." One of these judgments is necessarily true, the other is necessarily false.

This law can be written using the disjunction: p v ~ p, where P is any statement, 1 p is the negation of the statement p.

true (as indicated by the law of non-contradiction), but also false at the same time: if one of them is false, then the other must be true, the third is not given.

Law of sufficient reason. Our thoughts about any fact, phenomenon, event can be true or false.

Expressing a true thought, we must substantiate its truth, i.e. prove its validity. So, when bringing charges against the defendant, the accuser must provide the necessary evidence, substantiate the truth of his assertion. Otherwise, the accusation will be unfounded.

The requirement of proof, the validity of a thought expresses the law of sufficient reason: every thought is recognized as true if it has a sufficient reason. If there is b, then there is also its base a.

Thanks to the development of scientific knowledge, man is increasingly using the experience of all mankind as the basis of his thoughts,

enshrined in the laws and axioms of science, in the principles and regulations that exist in any field of human activity.

The truth of laws, axioms has been confirmed by the practice of mankind and therefore does not need new confirmation. To confirm any particular case, it is not necessary to substantiate it with the help of personal experience. If, for example, we know the law of Archimedes (each body immersed in a liquid loses as much in its weight as the liquid displaced by it weighs), then there is no point in immersing an object in a liquid in order to find out how much it loses in weight. The law of Archimedes will be a sufficient basis for confirming any particular case.

Thus, any other, already verified and established thought, from which the truth of this thought necessarily follows, can be a sufficient basis for any thought.

If the truth of proposition a implies the truth of proposition b, then a will be the reason for b, and b the consequence of this reason.

The connection between the foundation and the effect is a reflection in thinking of objective, including cause-and-effect relationships, which are expressed in the fact that one phenomenon (cause) gives rise to another phenomenon (effect). However, this reflection is not direct. In some cases, the logical basis may coincide with the cause of the phenomenon (if, for example, the idea that the number of traffic accidents has increased is justified by pointing to the cause of this phenomenon - ice on the roads). But most often there is no such coincidence. The judgment "It has rained recently" can be substantiated by the judgment "The roofs of the houses are wet"; the trace of the protectors of automobile awls is a sufficient basis for the judgment "A car passed at this place." Meanwhile, wet roofs and a trail left by a car are not the cause, but the consequence of these phenomena. Therefore, the logical connection between the basis and the effect must be distinguished from the causal relationship.

Validity is the most important property of logical thinking. In all cases when we affirm something, convince others of something, we must prove our judgments, give sufficient reasons confirming the truth of our thoughts.

lei. This is the fundamental difference between scientific thinking and non-scientific thinking, which is characterized by lack of evidence, the ability to accept various positions and dogmas on faith. This is especially characteristic of religious thinking, which is based not on proof, but on faith.

The law of sufficient reason is incompatible with various prejudices and superstitions. For example, there are ridiculous signs:

break a mirror - unfortunately, sprinkle salt - to a quarrel, etc., although there is no causal relationship between a broken mirror and misfortune, spilled salt and a quarrel. Logic is the enemy of superstition and prejudice. It requires the validity of judgments and is therefore incompatible with statements that are built according to the scheme "after this - therefore, because of this." This logical error occurs in cases where causality is confused with a simple sequence - in time, when the antecedent phenomenon is taken as the cause of the subsequent one.

Dmitry Alekseevich Gusev, Candidate of Philosophy, Associate Professor, Department of Philosophy, Moscow Pedagogical State University.

1. The law of identity

The first and most important law of logic is the law of identity, which was formulated by Aristotle in the treatise Metaphysics as follows: “... to have more than one meaning means not to have a single meaning; if words have no meaning, then all possibility of reasoning with each other, and in fact with oneself, is lost; for it is impossible to think of anything if one does not think of one thing. One could add to these words of Aristotle the well-known statement that to think (to speak) about everything means not to think (to speak) about nothing.

The law of identity states that any thought (any reasoning) must necessarily be equal (identical) to itself, that is, it must be clear, precise, simple, definite. In other words, this law prohibits confusion and substitution of concepts in reasoning (i.e., using the same word in different meanings or putting the same meaning in different words), creating ambiguity, evading the topic, etc. For example, the meaning of the phrase is not clear: “Because of absent-mindedness at tournaments, the chess player has repeatedly lost points.” Obviously, due to the violation of the law of identity, unclear statements (judgments) appear. The symbolic record of this law looks like this: a → a (read: “If a, then a”), where a is any concept, statement or whole reasoning.

When the law of identity is violated involuntarily, out of ignorance, then simply logical errors arise; but when this law is deliberately violated, with the aim of confusing the interlocutor and proving to him some false thought, then not just errors appear, but sophisms. Thus, sophism is an outwardly correct proof of a false thought with the help of a deliberate violation of logical laws.

Here is an example of sophism: “Which is better: eternal bliss or a sandwich? Of course, eternal bliss. And what could be better than eternal bliss? Of course, nothing! But a sandwich is better than nothing, therefore, it is better than eternal bliss. Try to find the catch in this reasoning on your own, determine where and how the law of identity is violated in it, and expose this sophism.

Here is another sophism: “Let's ask our interlocutor: “Do you agree that if you have lost something, then you don’t have it?” He replies, "I agree." Let’s ask him the second question: “Do you agree that if you didn’t lose something, then you have it?” “I agree,” he replies. Now let's give him the last and main question: "Did you lose your horns today?" What is left for him to answer? “I didn’t lose,” he says. “Therefore,” we say triumphantly, “you have them, because you yourself admitted at the beginning that if you didn’t lose something, then you have it.” Try to expose this sophism as well, to determine where and how the law of identity is violated in this outwardly correct reasoning.

However, not only vague judgments and sophisms are built on violations of the law of identity. By violating this law, you can create some kind of comic effect. For example, Nikolai Vasilyevich Gogol in the poem " Dead Souls”, Describing the landowner Nozdrev, he says that he was a “historical person”, because wherever he appeared, some kind of “story” necessarily happened to him. Many comic aphorisms are based on the violation of the law of identity. For example: "Do not stand anywhere, otherwise it will fall." Also, with the help of violation of this law, many jokes are created. For example:

“I broke my arm in two places.

Don't go to these places again.

As you can see, in all the examples given, the same technique is used: different meanings, situations, topics are mixed in identical words, one of which is not equal to the other, i.e., the law of identity is violated.

Violation of this law also underlies many of the problems and puzzles known to us since childhood. For example, we ask the interlocutor: “What (why) is there water in a glass beaker?” - deliberately creating ambiguity in this matter (why - for what and for what - for what subject, where). The interlocutor answers one question, for example, he says: "To drink, water the flowers", and we mean another question and, accordingly, another answer: "Behind the glass."

At the heart of all tricks is also a violation of the law of identity. The effect of any trick is that the magician does one thing, and the audience thinks completely different, that is, what the magician does is not equal (not identical) to what the audience thinks, which is why it seems that the magician is doing something unusual and mysterious. When opening the focus, we are usually visited by bewilderment and annoyance: it was so simple, how did we not notice it in time.

2. The law of contradiction

The law of contradiction says that if one judgment affirms something, and another denies the same thing about the same object, at the same time and in the same respect, then they cannot be true at the same time. For example, two judgments: “Socrates is high”, “Socrates is low” (one of them affirms something, and the other denies the same thing, because high is not low, and vice versa), cannot be simultaneously true when it comes to one and the same Socrates, at the same time of his life and in the same respect, that is, if Socrates is compared in growth not with different people at the same time, but with one person. It is clear that when we are talking about two different Socrates or about one Socrates, but at different times of his life, for example, at 10 years old and at 20 years old, or the same Socrates and at the same time of his life is considered in different ways. , for example, he is compared simultaneously with high Plato and low Aristotle, then two opposite judgments may well be true at the same time, and the law of contradiction is not violated. Symbolically, it is expressed by the following identically true formula: ¬ (a Λ ¬ a), (read: “It is not true that a and not a”), where a is some kind of statement.

In other words, the logical law of contradiction forbids asserting something and denying the same thing at the same time. But is it really possible for someone to assert something and then immediately deny the same thing? Will anyone seriously prove, for example, that one and the same person is both tall and short at the same time and in the same respect, or that he is both fat and thin; and blond, and brunette, etc.? Of course not. If the principle of the consistency of thinking is so simple and obvious, then is it worth calling it a logical law and generally paying attention to it?

The fact is that contradictions are contact, when the same thing is affirmed and immediately denied (the subsequent phrase denies the previous one in speech, or the subsequent sentence denies the previous one in the text) and distant, when there is a significant interval between contradictory judgments in speech or in the text. For example, at the beginning of his speech, the lecturer can put forward one idea, and at the end express a thought that contradicts it; so in a book, in one paragraph, what is denied in another can be affirmed. It is clear that contact contradictions, being too noticeable, almost never occur in thinking and speech. The situation is different with distant contradictions: being non-obvious and not very noticeable, they often pass by the visual or mental gaze, are involuntarily skipped, and therefore they can often be found in intellectual and speech practice. So, Vitaly Ivanovich Svintsov gives an example from one study guide, in which, with an interval of several pages, it was first stated: “In the first period of his work, Mayakovsky was no different from the Futurists,” and then: “Already from the very beginning of his work, Mayakovsky possessed qualities that significantly distinguished him from the representatives of Futurism.”

Contradictions can also be explicit and implicit. In the first case, one thought directly contradicts another, and in the second case, the contradiction follows from the context: it is not formulated, but implied. For example, in the textbook "Concepts of Modern Natural Science" (this subject is now being studied in all universities), from the chapter on Albert Einstein's theory of relativity, it follows that, according to modern scientific ideas, space, time and matter do not exist without each other: without one there is no another. And in the chapter on the origin of the Universe, it is said that it appeared about 20 billion years ago as a result of the Big Bang, during which matter was born, filling all space with itself. It follows from this statement that space existed before the appearance of matter, although in the previous chapter it was said that space cannot exist without matter. Explicit contradictions, as well as contact ones, are rare. Implicit contradictions, like distant ones, on the contrary, due to their invisibility, are much more common in thinking and speech.

An example of a contact and obvious contradiction is the following statement: “Driver N. grossly violated the rules when leaving the parking lot, because he did not take oral permission in writing.” Another example of a contact and obvious contradiction: “A young girl of advanced years with short hedgehog dark curly blond hair with a graceful gait of a gymnast, limping, entered the stage. Such contradictions are so obvious that they can only be used to create some kind of comic effect. Therefore, our task is to be able to recognize and eliminate them. An example of a contact and implicit contradiction: “This manuscript made on paper was created in Ancient Rus' in the 11th century. (in the 11th century there was no paper in Rus' yet)”.

Finally, probably each of us is familiar with the situation when we say to our interlocutor, or he tells us: "You contradict yourself." As a rule, in this case we are talking about distant or implicit contradictions, which, as we have seen, are quite common in various spheres of thought and life. Therefore, a simple and even primitive, at first glance, the principle of consistency of thinking has the status of an important logical law.